Homoclinic-Doubling Cascades

Cascades of period-doubling bifurcations have attracted much interest from researchers of dynamical systems in the past two decades as they are one of the routes to onset of chaos. In this paper we consider routes to onset of chaos involving homoclinic-doubling bifurcations. We show the existence of cascades of homoclinic-doubling bifurcations which occur persistently in two-parameter families of vector fields on ℝ3. The cascades are found in an unfolding of a codimension-three homoclinic bifurcation which occur an orbit-flip at resonant eigenvalues. We develop a continuation theory for homoclinic orbits in order to follow homoclinic orbits through infinitely many homoclinic-doubling bifurcations.

[1]  Floris Takens,et al.  Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations : fractal dimensions and infinitely many attractors , 1993 .

[2]  M. Peixoto,et al.  On an approximation theorem of Kupka and Smale , 1967 .

[3]  Eiji Yanagida,et al.  Branching of double pulse solutions from single pulse solutions in nerve axon equations , 1987 .

[4]  Martin Krupa,et al.  The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit , 1994, Ergodic Theory and Dynamical Systems.

[5]  Leonid P Shilnikov,et al.  ON SYSTEMS WITH A SADDLE-FOCUS HOMOCLINIC CURVE , 1987 .

[6]  Pavol Brunovský.  On one-parameter families of diffeomorphisms , 1970 .

[7]  P. Bonckaert.  On the Continuous Dependence of the Smooth Change of Coordinates in Parametrized Normal Form Theorems , 1993 .

[8]  James A. Yorke,et al.  Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation , 1982 .

[9]  Bo Deng,et al.  The Sil'nikov problem, exponential expansion, strong λ-lemma, C1-linearization, and homoclinic bifurcation , 1989 .

[10]  M. Komuro,et al.  MULTIPLE HOMOCLINIC BIFURCATIONS FROM ORBIT-FLIP I: SUCCESSIVE HOMOCLINIC DOUBLINGS , 1996 .

[11]  Hiroshi Kokubu,et al.  Bifurcations toN-homoclinic orbits andN-periodic orbits in vector fields , 1993 .

[12]  Raymond Kapral,et al.  Bifurcation phenomena near homoclinic systems: A two-parameter analysis , 1984 .

[13]  Alan R. Champneys,et al.  Death of period-doublings: locating the homoclinic-doubling cascade , 2000 .

[14]  Björn Sandstede,et al.  Constructing dynamical systems having homoclinic bifurcation points of codimension two , 1997 .

[15]  Pierre Collet,et al.  Period doubling bifurcations for families of maps on ℝn , 1981 .

[16]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[17]  Bernd Krauskopf,et al.  Resonant Homoclinic Flip Bifurcations , 2000 .

[18]  Björn Sandstede,et al.  Center Manifolds for Homoclinic Solutions , 2000 .

[19]  Nicholas C. Metropolis,et al.  On Finite Limit Sets for Transformations on the Unit Interval , 1973, J. Comb. Theory A.

[20]  Colin Sparrow,et al.  Local and global behavior near homoclinic orbits , 1984 .

[21]  Andrey Shilnikov,et al.  On bifurcations of the Lorenz attractor in the Shimizu-Morioka model , 1993 .

[22]  J. Yorke,et al.  An index for the global continuation of relatively isolated sets of periodic orbits , 1983 .

[23]  Marek Rychlik,et al.  Lorenz attractors through Šil'nikov-type bifurcation. Part I , 1990, Ergodic Theory and Dynamical Systems.

[24]  Ale Jan Homburg,et al.  Some global aspects of homoclinic bifurcations of vector fields , 1996 .

[25]  V. Naudot,et al.  The existence of infinitely many homoclinic doubling bifurcations from some codimension 3 homoclinic orbits , 1997 .

[26]  M. Feigenbaum.  Quantitative universality for a class of nonlinear transformations , 1978 .

[27]  W. D. Melo,et al.  ONE-DIMENSIONAL DYNAMICS , 2013 .

[28]  A. J. Homburg,et al.  Universal Scalings in Homoclinic Doubling Cascades , 2001 .

[29]  H. Epstein.  New proofs of the existence of the Feigenbaum functions , 1986 .

[30]  Exponential expansion with Šil'nikov's saddle-focus , 1989 .

[31]  Alan R. Champneys,et al.  Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations , 2001 .

[32]  Shlomo Sternberg,et al.  On the Structure of Local Homeomorphisms of Euclidean n-Space, II , 1958 .

[33]  The periodic points of renormalization , 1996, math/9604235.

[34]  V. V. Bykov,et al.  The bifurcations of separatrix contours and chaos , 1993 .

[35]  James A. Yorke,et al.  Families of periodic orbits: Virtual periods and global continuability , 1984 .

[36]  Pierre Coullet,et al.  ITÉRATIONS D'ENDOMORPHISMES ET GROUPE DE RENORMALISATION , 1978 .

[37]  Jorge Sotomayor,et al.  Generic Bifurcations of Dynamical Systems , 1973 .

[38]  S. Chow,et al.  Homoclinic bifurcation at resonant eigenvalues , 1990 .

[39]  Vincent Naudot.  Strange attractor in the unfolding of an inclination-flip homoclinic orbit , 1996 .

[40]  W. Kyner.  Invariant Manifolds , 1961 .

[41]  J. Yorke,et al.  Cascades of period-doubling bifurcations: A prerequisite for horseshoes , 1983 .

[42]  G. Belitskii.  Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class , 1973 .

[43]  J. Yorke,et al.  On the continuability of periodic orbits of parametrized three-dimensional differential equations☆ , 1983 .

[44]  Pierre Collet,et al.  Universal properties of maps on an interval , 1980 .